Algebra Examples

$\frac{3}{x^{2} - x - 6} \cdot \frac{x^{2} - 12 x + 27}{x^{2} + x} \div \frac{x - 9}{x + 2}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{3}{x^{2} - x - 6} \cdot \frac{x^{2} - 12 x + 27}{x^{2} + x} \frac{x + 2}{x - 9}$

Step 2

Factor $x^{2} - x - 6$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 2.2

Write the factored form using these integers.

$\frac{3}{(x - 3) (x + 2)} \cdot \frac{x^{2} - 12 x + 27}{x^{2} + x} \frac{x + 2}{x - 9}$

Step 3

Factor $x^{2} - 12 x + 27$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $27$ and whose sum is $- 12$ .

$- 9, - 3$

Step 3.2

Write the factored form using these integers.

$\frac{3}{(x - 3) (x + 2)} \cdot \frac{(x - 9) (x - 3)}{x^{2} + x} \frac{x + 2}{x - 9}$

Step 4

Simplify terms.

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Step 4.1

Factor $x$ out of $x^{2} + x$ .

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Step 4.1.1

Factor $x$ out of $x^{2}$ .

$\frac{3}{(x - 3) (x + 2)} \cdot \frac{(x - 9) (x - 3)}{x \cdot x + x} \frac{x + 2}{x - 9}$

Step 4.1.2

Raise $x$ to the power of $1$ .

$\frac{3}{(x - 3) (x + 2)} \cdot \frac{(x - 9) (x - 3)}{x \cdot x + x^{1}} \frac{x + 2}{x - 9}$

Step 4.1.3

Factor $x$ out of $x^{1}$ .

$\frac{3}{(x - 3) (x + 2)} \cdot \frac{(x - 9) (x - 3)}{x \cdot x + x \cdot 1} \frac{x + 2}{x - 9}$

Step 4.1.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

$\frac{3}{(x - 3) (x + 2)} \cdot \frac{(x - 9) (x - 3)}{x (x + 1)} \frac{x + 2}{x - 9}$

Step 4.2

Cancel the common factor of $x - 3$ .

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Step 4.2.1

Factor $x - 3$ out of $(x - 9) (x - 3)$ .

$\frac{3}{(x - 3) (x + 2)} \cdot \frac{(x - 3) (x - 9)}{x (x + 1)} \frac{x + 2}{x - 9}$

Step 4.2.2

Cancel the common factor.

$\frac{3}{(x - 3) (x + 2)} \cdot \frac{(x - 3) (x - 9)}{x (x + 1)} \frac{x + 2}{x - 9}$

Step 4.2.3

Rewrite the expression.

$\frac{3}{x + 2} \cdot \frac{x - 9}{x (x + 1)} \frac{x + 2}{x - 9}$

Step 4.3

Multiply $\frac{3}{x + 2}$ by $\frac{x - 9}{x (x + 1)}$ .

$\frac{3 (x - 9)}{(x + 2) (x (x + 1))} \cdot \frac{x + 2}{x - 9}$

Step 4.4

Cancel the common factor of $x - 9$ .

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Step 4.4.1

Factor $x - 9$ out of $3 (x - 9)$ .

$\frac{(x - 9) \cdot 3}{(x + 2) (x (x + 1))} \cdot \frac{x + 2}{x - 9}$

Step 4.4.2

Cancel the common factor.

$\frac{(x - 9) \cdot 3}{(x + 2) (x (x + 1))} \cdot \frac{x + 2}{x - 9}$

Step 4.4.3

Rewrite the expression.

$\frac{3}{(x + 2) (x (x + 1))} (x + 2)$

Step 4.5

Cancel the common factor of $x + 2$ .

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Step 4.5.1

Cancel the common factor.

$\frac{3}{(x + 2) (x (x + 1))} (x + 2)$

Step 4.5.2

Rewrite the expression.

$\frac{3}{x (x + 1)}$